To find [tex]\((B \cap A)^{\prime} \cup C\)[/tex], we can follow the step-by-step approach:
1. Find the Intersection [tex]\(B \cap A\)[/tex]:
[tex]\(B = \{f, h, p, x, z\}\)[/tex]
[tex]\(A = \{f, r, x, y, z\}\)[/tex]
The intersection of [tex]\(B\)[/tex] and [tex]\(A\)[/tex] (elements common to both [tex]\(B\)[/tex] and [tex]\(A\)[/tex]) is:
[tex]\[B \cap A = \{f, x, z\}\][/tex]
2. Find the Complement of [tex]\(B \cap A\)[/tex] with respect to the Universal Set [tex]\(U\)[/tex]:
[tex]\(U = \{f, g, h, p, q, r, x, y, z\}\)[/tex]
The complement of [tex]\(B \cap A\)[/tex], denoted as [tex]\((B \cap A)^{\prime}\)[/tex], is the set of elements in [tex]\(U\)[/tex] that are not in [tex]\(B \cap A\)[/tex]:
[tex]\[(B \cap A)^{\prime} = U - (B \cap A) = \{f, g, h, p, q, r, x, y, z\} - \{f, x, z\}\][/tex]
So, the complement set is:
[tex]\[(B \cap A)^{\prime} = \{r, g, q, y, h, p\}\][/tex]
3. Find the Union of [tex]\((B \cap A)^{\prime}\)[/tex] and [tex]\(C\)[/tex]:
[tex]\(C = \{p, q, r, x\}\)[/tex]
The union of [tex]\((B \cap A)^{\prime}\)[/tex] and [tex]\(C\)[/tex] is the set of elements that are in either [tex]\((B \cap A)^{\prime}\)[/tex] or [tex]\(C\)[/tex] (or in both):
[tex]\[(B \cap A)^{\prime} \cup C = \{r, g, q, y, h, p\} \cup \{p, q, r, x\}\][/tex]
Combining these sets, we get:
[tex]\[(B \cap A)^{\prime} \cup C = \{r, g, q, y, h, p, x\}\][/tex]
Therefore, the final answer in roster form is:
[tex]\[
(B \cap A)^{\prime} \cup C = \{r, g, x, q, y, h, p\}
\][/tex]
